Question: Solve the exponential equation for $x$. 64 7 x − 1 4 2 x + 3 = 4 9 x − 4 \dfrac{64\^{ 7x-1}}{4\^{ 2x+3}}=4\^{ 9x-4} $x=$
The strategy Let's write $64$ in base $4$. Then, using the properties of exponents, we can express the entire left hand side of the equation as $4$ raised to some linear function. Finally, we can equate the exponents of the resulting equation to solve for the unknown. Simplifying the left hand side 64 7 x − 1 4 2 x + 3 = ( 4 3 ) 7 x − 1 4 2 x + 3 = 4 21 x − 3 4 2 x + 3 = 4 21 x − 3 − ( 2 x + 3 ) = 4 19 x − 6 ( 64 = 4 3 ) ( ( a n ) m = a n ⋅ m ) ( a n a m = a n − m ) \begin{aligned}\dfrac{64\^{ 7x-1}}{4\^{ 2x+3}}&=\dfrac{(4^3)\^{ 7x-1}}{4\^{ 2x+3}}&&&&(64=4^3) \\\\\\\\ &=\dfrac{4\^{ C{21x-3}}}{4\^{ {2x+3}}} &&&&((a^n)^m=a^{n\cdot m})\\\\\\\\ &=4\^{ C{21x-3} \ - \ ({2x+3})}&&&&(\dfrac{a^n}{a^m}=a^{n-m})\\\\\\\\ &=4\^{ 19x-6} \end{aligned} Solving the equation We obtain the following equation. 4 19 x − 6 = 4 9 x − 4 4\^{ 19x-6}=4\^{ 9x-4} Now we can equate the exponents and solve for $x$. $\begin{aligned} 19x-6 &=9x-4\\\\ x &= \dfrac{1}{5}\end{aligned}$ The answer The answer is $x=\dfrac{1}{5}$. You can check this answer by substituting $\it{x=\dfrac{1}{5}}$ in the original equation and evaluating both sides.